Question

The times between $7$ and $8$ o’clock, correct to the nearest minute, when the hands of a clock will form an angle of $84$ degrees are:

$7{:}23$ and $7{:}53$

$7{:}20$ and $7{:}50$

$7{:}22$ and $7{:}53$

$7{:}23$ and $7{:}52$

$7{:}21$ and $7{:}49$

Answer: A

Concepts:clock linear equation estimation

Difficulty rating: 1870

Small Hint:

At $t$ minutes after $7{:}00,$ the signed separation of the hands is $210^\circ-5.5t^\circ$

Big Hint:

Solve $\lvert210-5.5t\rvert=84$ for both values of $t$

Solution:

At $t$ minutes after $7{:}00,$ the hour hand is at $210^\circ+0.5t^\circ$ and the minute hand is at $6t^\circ.$ Thus $|210-5.5t|=84.$ The two solutions are $\begin{aligned} t&=\frac{126}{5.5}\approx22.91,\\ t&=\frac{294}{5.5}\approx53.45. \end{aligned}$ To the nearest minute, the times are $7{:}23$ and $7{:}53.$

Thus, the correct answer is A.

1954 AMC 12 Problem 50

Answer: A

Difficulty rating: 1870

Small Hint:

Big Hint:

Solution:

Problem 50 in Other Years